Applications - A Guide to MATLAB for Beginners and Experienced Users - page 212

solution curves to the system (4) for initial data
x (0) = 0:1 / 12:13 / 12
y (0) = 0:1 / 12:13 / 12 .
( Hint : Use axis to limit your view to the square 0 ≤ x , y ≤ 13 / 12.)
(d) The picture you drew is called a phase portrait of the system. In-
terpret it. Explain the long-term behavior of any population distri-
bution that starts with only one species present. Relate it to part
(b). What happens in the long term if both populations are present
initially? Is there an initial population distribution that remains
undisturbed? What is it? Relate those numbers to the model (4).
(e) Now replace 0 . 5 in the model by 2; that is, consider the new model
x ( t ) = x − x 2

− 2 xy

(5)

˙ y ( t ) = y − y 2

− 2 xy .
Draw the phase portrait. (Use the same initial data and viewing
square.) Answer the same questions as in part (d). Do you see a
special solution trajectory that emanates from near the origin and
proceeds to the special fixed point? And another trajectory from the
upper right to the fixed point? What happens to all population dis-
tributions that do not start on these trajectories?
(f) Explain why model (4) is called “peaceful coexistence” and model (5)
is called “doomsday.” Now explain heuristically why the coefficient
change from 0.5 to 2 converts coexistence into doomsday.
12. Build a SIMULINK model corresponding to the pendulum equation
x ( t ) =− 0 . 5 x ( t ) − 9 . 81sin( x ( t )) (6)

from The 360˚ Pendulum in Chapter 9. You will need the Trigonometric
Function block from the Math library. Use your model to redraw some of
the phase portraits.
13. As you know, Galileo and Newton discovered that all bodies near the
earth's surface fall with the same acceleration g due to gravity, approx-
imately 32.2 ft/sec 2 . However, real bodies are also subjected to forces due
to air resistance. If we take bothgravity and air resistance into account,
a moving ball can be modeled by the differential equation
x = [0 , − g ] − c x x . (7)

Here x , a function of the time t , is the vector giving the position of the ball
(the first coordinate is measured horizontally, the second one vertically),
x is the velocity vector of the ball, x is the acceleration of the ball, x